具有粗糙初值的Landau-Lifshitz-Gilbert方程的整体解的存在性

基金项目:

关于液晶和铁磁物体中某些方程的数学研究;华南理工大学中央高校科研业务费项目

详细信息

Global existence of Landau-Lifshitz-Gilbert system with rough initial data

摘要: 关注高维Landau-Lifshitz-Gilbert方程的整体解的存在性问题,证明了当初始值半范数 $[Z_0]_{BMO(R^n)}$ 充分小时,Landau-Lifshitz-Gilbert方程柯西问题存在整体解: 通过球面投射的方法, 把Landau-Lifshitz-Gilbert方程转化为一个非线性Schr\"odinger方程,然后研究该方程的整体解的存在性,最后通过逆运算,得到原来方程的解的整体存在性.
- 球面投射.
Abstract: The global solutions to Landau-Lifshitz-Gilbert equation in high dimensions are considered. The global well-posedness of the Cauchy problem of the Landau-Lifshitz-Gilbert equation in $R^n$ for any initial data $Z_0\in S^2$ with small $[Z_0]_{BMO(R^n)} is established.$ The method is based on priori estimates of a nonlinear Schr\"odinger equation obtained from the Landau-Lifshitz-Gilbert equation by the stereographic projection.
- Stereographic Projection